What is black hole entropy?

Two important thermodynamic quantities are temperature
and entropy. Temperature we all know from our fevers,
weather reports and ovens. Entropy however is foreign to everyday life
for most people.
Suppose we have a box filled with gas
of some type of molecule called M. The temperature of that gas in that
box tells us the average kinetic energy of those vibrating molecules
of gas. Each molecule as a quantum particle has quantized energy states,
and if we understand the quantum theory of those molecules, theorists
can count up the available quantum microstates of those
molecules and get some number. The entropy is the logarithm
of that number.
When it was discovered that black holes
can decay by quantum processes, it was also discovered that
black holes seem to have the thermodynamic properties of temperature
and entropy. The temperature of the black hole is inversely proportional
to its mass, so the black hole gets hotter and hotter as it decays.
The entropy of a black hole is one fourth of
the area of the event horizon, so the entropy gets smaller and smaller
as the black hole decays and the event horizon area becomes smaller
and smaller.
But until string theory there was not a clear
relation between quantum microstates of a quantum theory and this supposed
black hole entropy.

Black holes and branes in string theory

A black hole is an object that is described
by a spacetime geometry that is a solution to the Einstein equation.
In string theory at large distance scales, solutions to the Einstein
equation are only modified by very small corrections. But it has been
discovered through string duality relations that spacetime
geometry is not a fundamental concept in string theory, and at
small distance scales or when the forces are very strong, there is an
alternate description of the same physical system that appears to be
very different.
A special type of black hole that is very important
in string theory is called a BPS black hole.
A BPS black hole has both charge (electric and/or magnetic) and mass,
and the mass and the charges satisfy an equality that leads to unbroken
supersymmetry in the spacetime near the black hole. This supersymmetry
is very important because it results in the disappearance of messy quantum
corrections, so that precise answers about the physics near the black
hole horizon can be found by simple calculations.
In the previous section we learned that string
theories contain objects called p-branes
and D-branes. Since a point
can be thought of as a zero-brane, a
natural generalization of a black hole is a black
p-brane. And there are also BPS black
p-branes.
But there's also a relationship between black
p-branes and D-branes. At large values of the charge, spacetime geometry
is a good description of of a black p-brane system. But when the charge
is small, the system can be described by a
bunch of weakly interacting D-branes.
In this weakly coupled D-brane limit, with the
BPS condition satisfied, it is possible to calculate the number of available
quantum states. This answer depends on the charges of the D-branes in
the system.
When we go back to the geometrical limit of
the equivalent black hole of p-brane system with the same charges and
masses, we find that the entropy of the D-brane
system matches the entropy as calculated from the black hole or p-brane
event horizon area.
This was a fantastic result for string theory.
But can we now say that D-branes provide the fundamental quantum microstates
of a black hole that underlie black hole thermodynamics? The D-brane
calculation is only easily performed for the supersymmetric BPS black
objects. Most black holes in the Universe probably have very little
if any electric or magnetic charge, and are very far from being BPS
objects. It's still a challenge to compute the black hole entropy for
such an object using D-branes.